The long term objective of this research is to develop a method to infer and image tissue hardness using ultrasounds. This method would bring new quantitative informations about the condition of a tissue, similar to what palpation does qualitatively when sensing "hard" tumor or a muscle under contraction. It consists essentially in measuring the tissue deformation induced through an external force, the deformation being large in a compliant (i.e. soft) material, and smaller in a rigid (hard) one. As proposed here, tissue deformation is assessed by tracking the spatio- temporal changes induced by the compression force on the ultrasound speckle patterns. Technically, this application is for developing a model-based approach to compute the elasticity distribution (also called the elastogram) using ultrasound speckle displacement data. First it is proposed to study a direct elasticity problem; here the aim is to provide an analytical framework and a computer model of image formation for speckle correlation and motion, assuming an elastic scattering medium subjected to deformations. From a theoretical point of view, this will serve to better understand the complex relationship between the spatio-temporal changes in ultrasound signals and the underlying tissue motion. From a practical standpoint, this will serve to develop and test tissue motion estimators based either on speckle correlation measurements or on optical flow. Tissue motion estimation is the first step in solving the inverse elasticity problem which can be stated as: determining the spatial distribution of elasticity that could best reproduce the estimated tissue motion under constraints provided by the elasticity equations and boundary conditions. It is proposed to develop an iterative procedure to seek the optimal elastogram, i.e. an optical flow algorithm adapted to this problem; this elastogram will be compared to the one which is currently studied by others, and which uses correlation-based one-dimensional tissue motion estimator. It is finally proposed to study the improvement on the estimation of elasticity distribution through use of 2D and 3D speckle motion.